Definition A quadratic inequality can be written in the form ax2 bx c 0 or ax2 bx c 0 where a, b, and c are real numbers and a 0. ( and may be substituted for and .) 1 Solve a Quadratic Inequality by Graphing To understand how to solve a quadratic inequality, let’s look at the graph of a quadratic function. EXAMPLE 1 a) Graph y x2 2x 3. b) Solve x2 2x 3 0. c) Solve x2 2x 3 0. Solution a) The graph of the quadratic function y x2 2x 3 is a parabola that opens upward. Use the vertex formula to confi rm that the vertex is (1, 4). To fi nd the y-intercept, let x 0 and solve for y. y 02 2(0) 3 y 3 The y-intercept is (0, 3). To fi nd the x-intercepts, let y 0 and solve for x. 0 x2 2x 3 0 (x 3)(x 1) Factor. x 3 0 or x 1 0 Set each factor equal to 0. x 3 or x 1 Solve. b) We will use the graph of y x2 2x 3 to solve the inequality x2 2x 3 0. That is, to solve x2 2x 3 0 we must ask ourselves, “Where are the y-values of the function less than zero?” The y-values of the function are less than zero when the x-values are greater than 1 and less than 3, as shown to the right. y 5 5 5 5 y 5 1 x 5 5 y y 5 The solution set of x2 2x 3 0 (in interval notation) is (1, 3). Be sure to notice the difference between and as well as and . x y x2 2x x www.mhhe.com/messersmith SECTION 10.7 Quadratic and Rational Inequalities 681
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